Product of normal PDFs

October 29, 2012
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(This article was originally published at The Endeavour » Statistics, and syndicated at StatsBlogs.)

The product of two normal PDFs is proportional to a normal PDF. This is well known in Bayesian statistics because a normal likelihood times a normal prior gives a normal posterior. But because Bayesian applications don’t usually need to know the proportionality constant, it’s a little hard to find. I needed to calculate this constant, so I’m recording the result here for my future reference and for anyone else who might find it useful.

Denote the normal PDF by

phi(x; m, s) = frac{1}{sqrt{2pi} s} expleft(-frac{(x-m)^2}{2s^2}right)

Then the product of two normal PDFs is given by the equation

phi(x; mu_1, sigma_1) , phi(x; mu_2, sigma_2) = phi(mu_1; mu_2, sqrt{sigma_1^2 + sigma_2^2}) ,phi(x, mu, sigma)

where

mu = frac{ sigma_1^{-2} mu_1 + sigma_2^{-2} mu_2}{sigma_1^{-2} + sigma_2^{-2} }

and

sigma^2 = frac{sigma_1^2 sigma_2^2}{sigma_1^2 + sigma_2^2}

Note that the product of two normal random variables is not normal, but the product of their PDFs is proportional to the PDF of another normal.



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